Enter Number
Given S = 20:
Calculate:√20 using the Newtons Method
Build Newtons Method
The square root of a number can be represented
ƒ(x) = x2 - S
Take the Derivative of this
ƒ'(x) = 2x
Since the square root > 0, start with x0 = 1
Calculate x1
x1 = x0 + (ƒ(x0) - S)/ƒ'(x0)
x1 = 1 + (12 - 20)/2(1)
x1 = 1 + (1 - 20)/2
x1 =1 + -19/2
x1 = 1 + -9.5
x1 = 10.5
Calculate x2
x2 = x1 + (ƒ(x1) - S)/ƒ'(x1)
x2 = 10.5 + (10.52 - 20)/2(10.5)
x2 = 10.5 + (110.25 - 20)/21
x2 =10.5 + 90.25/21
x2 = 10.5 + 4.297619047619
x2 = 6.202380952381
Calculate x3
x3 = x2 + (ƒ(x2) - S)/ƒ'(x2)
x3 = 6.202380952381 + (6.2023809523812 - 20)/2(6.202380952381)
x3 = 6.202380952381 + (38.469529478458 - 20)/12.404761904762
x3 =6.202380952381 + 18.469529478458/12.404761904762
x3 = 6.202380952381 + 1.4889064070926
x3 = 4.7134745452884
Calculate x4
x4 = x3 + (ƒ(x3) - S)/ƒ'(x3)
x4 = 4.7134745452884 + (4.71347454528842 - 20)/2(4.7134745452884)
x4 = 4.7134745452884 + (22.216842289081 - 20)/9.4269490905767
x4 =4.7134745452884 + 2.2168422890814/9.4269490905767
x4 = 4.7134745452884 + 0.23516009981398
x4 = 4.4783144454744
Calculate x5
x5 = x4 + (ƒ(x4) - S)/ƒ'(x4)
x5 = 4.4783144454744 + (4.47831444547442 - 20)/2(4.4783144454744)
x5 = 4.4783144454744 + (20.055300272545 - 20)/8.9566288909488
x5 =4.4783144454744 + 0.055300272544521/8.9566288909488
x5 = 4.4783144454744 + 0.0061742284086824
x5 = 4.4721402170657
Final Answer
x5 = 4.4721402170657
You have 1 free calculations remaining
What is the Answer?
x5 = 4.4721402170657
How does the Newton Method Calculator work?
Free Newton Method Calculator - Calculates the square root of a positive integer using the Newton Method
This calculator has 1 input.
What 3 formulas are used for the Newton Method Calculator?
ƒ(x) = x2 - Sƒ'(x) = 2x
xn = xn - 1 + (ƒ(xn - 1) - S)/ƒ'(xn - 1)
For more math formulas, check out our Formula Dossier
What 3 concepts are covered in the Newton Method Calculator?
algorithmA process to solve a problem in a set amount of timenewtons methodanother numerical method for solving an equation f...square roota factor of a number that, when multiplied by itself, gives the original number√x
Example calculations for the Newton Method Calculator
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